Smooth:

A mapping f of a surface into space is smooth provided
it has sufficiently many partial derivatives, and that the mixed
partials do not depend on the order of differentiation. Here
"sufficiently many" depends on the context. Usually 2 is sufficient,
but often smooth means infinitely many.

The basic idea is that the surface admits a differentiable structure,
so that it is possible to do calculus on the surface. For most
points, there will be a well-defined tangent plane, though this does
not rule out self-intersection and some other degenerate behavior.